Zheng et al.
Vol. 19, No. 4 / April 2002 / J. Opt. Soc. Am. B
839
Femtosecond second-harmonic generation in
periodically poled lithium niobate
waveguides with simultaneous strong
pump depletion and group-velocity walk-off
Zheng Zheng* and Andrew M. Weiner
School of Electrical and Computer Engineering, Purdue University, 1285 EE Building, West Lafayette,
Indiana 47907-1285
Krishnan R. Parameswaran, Ming-Hsien Chou,† and Martin M. Fejer
E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305-4085
Received May 15, 2001; revised manuscript received August 1, 2001
We report studies of second-harmonic generation (SHG) of femtosecond pulses in long periodically poled
lithium niobate waveguides under large conversion conditions. Strong saturation of the SHG efficiency was
observed, accompanied by spectral and temporal distortion of the pump pulse. Our simulation studies suggest that the pulse distortions may be caused by the interaction of the phase-matched SHG process and an
additional cascaded ␹ (2) process or processes, leading to a large nonlinear phase modulation. Such additional
cascaded ␹ (2) processes could be caused by the existence of multiple transverse modes in the nonlinear waveguide. These phenomena, which to our knowledge have not been reported previously, may have a significant
effect on studies of high-power short-pulse parametric process in waveguide devices and on the design of novel
nonlinear optical waveguide devices for such applications. © 2002 Optical Society of America
OCIS codes: 190.2620, 190.4390, 190.7110, 230.4320.
1. INTRODUCTION
Second-harmonic generation (SHG), first studied in the
early 1960s, is among the best-known nonlinear optical
phenomena. Investigations of SHG in response to ultrafast optical pulses were also reported in the 1960s.1,2
These early investigations revealed the importance of
group-velocity mismatch (GVM) between the fundamental and the second-harmonic waves when the GVM is sufficiently large compared with the pulse duration, the
GVM leads to a broadening of the generated secondharmonic pulse with respect to the input pulse, together
with a narrowing of the phase-matching bandwidth. To
avoid these usually unwanted effects, femtosecond SHG
experiments normally use short nonlinear crystals with a
small GVM. However, we recently used thick nonlinear
crystals with a large GVM to perform several femtosecond
SHG experiments. Our results demonstrate several interesting and unique features. These include the possibility of high conversion efficiency at low input power,3 especially when waveguide devices are used,4 at the cost of
the second-harmonic pulse width, and also include alloptical signal processing for high-contrast ultrafast waveform recognition.4,5 Here we present detailed experimental data on femtosecond SHG in periodically poled
lithium niobate (PPLN) waveguides under conditions of
simultaneous strong pump depletion and large GVM.
This parameter regime has seen very little, if any, previous experimental investigation.
0740-3224/2002/040839-10$15.00
Nonlinear devices based on quasi-phase-matched
(QPM) materials such as PPLN6–8 can provide relatively
large nonlinearities while opening up additional degrees
of freedom for engineering the nonlinear output.9–11 For
example, nonlinear crystals with chirped and patterned
QPM gratings have been used for pulse compression and
pulse shaping in femtosecond SHG.9,10 Nonlinear optical
waveguides promise even greater nonlinear efficiencies
owing to their confined optical fields and long interaction
lengths. However, additional complexities arise in designing waveguide nonlinear devices. Issues related to
control and optimization of interacting spatial modes
have a significant effect on device behavior.12,13 Usually
waveguide design efforts focus on the pair of modes at
fundamental and second-harmonic wavelengths that give
the best mode overlap and thus the highest efficiency for
frequency conversion. Although in most cases other spatial modes exist, their effect on nonlinear device performance is often neglected.
Here we present the results of experimental and theoretical studies of femtosecond pulse SHG in long PPLN
waveguides under the high pump depletion condition.
Although a SHG conversion efficiency of ⬃60% is easily
obtained at input energies of only a few picojoules, we observed a surprisingly strong saturation that prevented
the efficiency from increasing significantly above this
value. We also observed a strong reshaping of the temporal and spectral profiles of the input pulse that is not
© 2002 Optical Society of America
840
J. Opt. Soc. Am. B / Vol. 19, No. 4 / April 2002
expected under the standard two-coupled-wave treatment
of SHG. Our theoretical simulation studies suggest that
an additional, very large nonlinear phase modulation process may be responsible for these effects. Although the
origin of this phase modulation is not yet pinned down,
we believe that new channels for cascaded ␹ ( 2 ) processes
involving multiple spatial waveguide modes may be responsible.
High depletion of ultrashort pulses through SHG may
find potential applications in wavelength conversion,
short-wavelength generation, generation of amplitudesqueezed light,14 and all-optical signal processing in optical communication systems. For example, an optical
add–drop multiplexer for ultrashort pulse code-division
multiple-access optical communications15 could be realized by the scheme illustrated in Fig. 1. The function of
the add–drop multiplexer is to selectively drop a channel
represented by a specific optical waveform (or code) from
the main data stream in the network without affecting
other transmitted channels represented by other distinct
waveforms (codes). We previously demonstrated that
SHG in a long nonlinear crystal with large GVM can function as a spectral phase correlator, which provides a
strong contrast (approaching 30 dB) in the SHG yield between certain properly and improperly coded
waveforms.5,4 Therefore a channel dropper could be constructed with two pulse shapers and a long SHG crystal,
as shown in Fig. 1. After the first pulse shaper performs
a decoding operation, the SHG process preferentially converts waveforms with the selected code into the second
harmonic, which can then be separated from the remaining data by a wavelength demultiplexer. The second
pulse shaper performs the operation inverse to that of the
first pulse shaper, so that the output waveforms in the remaining channels are the same as at the input. Note
that complete dropping of the selected channel would require nearly 100% conversion of the appropriate optical
pulses to the second-harmonic wavelength, which has not
been achieved in our studies so far. This provides a motivation for the research presented in this paper, in which
we seek to gain insight into the processes limiting the
achievement of 100% conversion in the large GVM regime.
In this paper we first present our experimental results
on SHG of femtosecond pulses in a long PPLN waveguide
at a 1.56-␮m wavelength under high-depletion conditions.
We observe strong saturation of the SHG conversion efficiency accompanied by temporal breakup of the 1.56-␮m
pump pulse and distortion of its power spectrum, which
cannot be explained by the SHG process alone. We then
present theoretical studies in which we solve the coupled
propagation equations numerically. By assuming a negative nonlinear phase modulation process approximately 2
orders of magnitude stronger than the normal optical
Kerr effect, we are able to simulate the experimental re-
Fig. 1.
Zheng et al.
sults. Finally, we discuss simulation results supporting
the possibility that this nonlinear phase distortion arises
from a cascaded ␹ ( 2 ) process that may involve additional
non-phase-matched waveguide modes.
2. EXPERIMENTAL STUDIES
In our experimental setup, the fundamental light came
from a Spectra-Physics femtosecond optical parametric
oscillator synchronously pumped by a mode-locked Ti:sapphire laser. The pulses were nearly transform limited
and had a width of ⬃200–250 fs. The spectral FWHM
bandwidth was ⬃12 nm, and the center wavelength was
tunable around 1560 nm. The beam was focused into the
PPLN waveguide by free-space coupling.
Our PPLN sample4 included a 55.5-mm-long SHG section with a 12-␮m waveguide width and with a nominally
uniform grating period of 14.75 ␮m. The 12-␮m waveguide width was chosen to achieve noncritical QPM, so
that the phase-matching frequency became relatively insensitive to small errors in the waveguide dimensions.12
The input and output ends of the sample incorporated
4-␮m-wide mode-filter sections, which were linearly tapered over a several-millimeter distance to the 12-␮m
width of the SHG region. Since the SHG section supports multiple transverse modes at both the pump and
the SHG wavelengths, the mode filter facilitates coupling
into the fundamental transverse mode of the waveguide
by suppressing higher-order spatial modes at the pump
wavelength.16 Waveguide fabrication was performed by
proton exchange to a depth of 0.71 ␮m followed by annealing for 26 h at 328 °C in air. The phase-matching condition was temperature tuned, and, for a 1560-nm input,
the phase-matching temperature was ⬃100 °C. The
GVM of the annealed proton exchanged waveguide is
⬃0.37 ps/mm, corresponding to a total GVM of ⬃20.5 ps.
This implies a theoretical second-harmonic phasematching bandwidth of ⬃0.1 nm.
We measured the SHG efficiency as a function of input
pump power, with the phase-matching wavelength of the
PPLN sample set for phase matching of an ⬃779-nm
second-harmonic wavelength (⬃1558 center wavelength
for the fundamental). As is shown in Fig. 2, we can see
an ⬃50%/pJ small-signal SHG efficiency for pump pulse
energies below 1 pJ, similar to what was observed in Ref.
4. However, the SHG efficiency quickly saturates at
⬃60% at pulse energy levels above 2 pJ. Also plotted in
the figure is the theoretical curve, calculated assuming an
ideal SHG process under the large GVM condition. The
calculation procedure is described in greater detail in Section 3. It is obvious that the experimental efficiency
saturates considerably more strongly than does the theorical.
We note that, under high depletion conditions (severalpicojoule pump energy), we also observed some weak
Scheme of proposed add–drop multiplexer for code-division multiple-access systems.
SH, second harmonic.
Zheng et al.
Fig. 2. SHG efficiency versus pump pulse energy. Solid curve,
experimental result; Dashed curve, theoretical result.
Vol. 19, No. 4 / April 2002 / J. Opt. Soc. Am. B
841
of the trace is ⬃250 fs, which is close to that estimated
from the inverse spectral bandwidth. This indicates that
there is no significant pulse broadening or distortion at
low power; accordingly, we conclude that the effect of
group-velocity dispersion (GVD) on the propagation of the
fundamental pulse is small in our waveguide device. On
the other hand, at high input levels (⬎2 pJ) the pump
pulse begins to change significantly. At an ⬃10-pJ input,
a double-hump pulse shape was observed, as shown in
Fig. 4(b).
We also repeated the cross-correlation measurements
with the phase-matching wavelength detuned to ⬃774
nm (⬃1548 nm at the fundamental). As a result, the
SHG output was reduced by more than 1 order of magnitude compared with the phase-matched cases discussed
above. For both low- and high-power inputs, the observed cross-correlation traces, shown in Fig. 5, are essentially identical to the low-power trace shown in Fig. 4(a).
Furthermore, the distortion evident at high power in the
phase-matched case, Fig. 4(b), is no longer observed.
This indicates that the pulse distortion is linked to the
parametric process(es) and does not arise solely from processes such as the optical Kerr effect (e.g., self-phase
modulation, SPM).
The spectra of the pump pulses exiting the SHG sample
were also measured under similar conditions with an op-
Fig. 3. Second-harmonic spectrum shape under low-power
pump conditions.
green light generated from the sample, which we attribute to sum-frequency mixing of the fundamental and
the second harmonic waves through a nearly phasematched third-order QPM process.17 We also measured
the output pump power and the second-harmonic power
individually as a function of input pump powers up to 10
pJ. The ratio of the total output power to the input
power remained constant to within our experimental accuracy of ⬃10%. This shows that nonlinear loss mechanisms do not play a significant role in our experiments.
To gain insight into the surprising saturation behavior,
we performed a series of additional measurements.
First, we measured the SHG spectrum by using an optical
spectrum analyzer, which had a spectral resolution of
⬃0.08 nm. As is seen in Fig. 3, at low input powers the
SHG spectrum had a FWHM bandwidth of ⬃0.3 nm with
a long oscillating tail on the short-wavelength side. The
deviation from the ideal sinc2 shape may be caused by inhomogeneities in the device structure or a nonuniform
temperature distribution in the PPLN oven.18 We address this issue in Section 3.
Second, the temporal profile of the transmitted pump
pulse at 1560 nm was measured by means of electric field
cross correlation. The reference pulse was taken directly
from the output of the optical parametric oscillator. The
data are shown in Fig. 4. At low input power the width
Fig. 4. Pump pulse temporal shape when it is phase matched
for SHG (a) at 0.5-pJ power level, (b) at 10-pJ power level.
Fig. 5. Pump pulse temporal shape when it is not phase
matched for SHG (a) at 0.5-pJ power level, (b) at 10-pJ power
level.
842
Fig. 6.
J. Opt. Soc. Am. B / Vol. 19, No. 4 / April 2002
Pump spectrum variation as the pump power increases.
Fig. 7. Output pump spectrum under high-power conditions.
The center wavelength is varied while the phase-matching wavelength of the sample is fixed at ⬃1558 nm. The labels show the
center wavelengths of the input pump spectrum.
Zheng et al.
short- and the long-wavelength tails of the spectrum. We
also measured the pump output spectra for a fixed
⬃1558.5-nm phase-matching wavelength and 10-pJ input
pump energy while tuning the input pump wavelength.
As is shown in Fig. 7, the results show significant but
asymmetric distortions at this energy level as long as the
phase-matching wavelength is within the FWHM of the
pump spectrum.
Besides the spectral intensity distributions, we further
analyzed the pump signal by performing spectral
interferometry19 measurements to retrieve the spectral
phase information. When the pump spectrum is centered at the phase-matching wavelength, ⬃1558.5 nm, a
significant phase jump centered at the spectral amplitude
dip begins to emerge as the pump power increases. As is
shown in Fig. 8, the phase variation exceeds 2 rad at a
10-pJ input energy level. The measurements performed
under low-power conditions showed no sign of such a
phase variation.
Summarizing this section, our experiments reveal a
surprisingly strong saturation of the SHG efficiency accompanied by a significant nonlinear distortion of the
pump pulses. Our experimental results also seem to exclude simple effects, such as GVD or SPM arising from
the optical Kerr effect, as the origin of these phenomena.
An explanation may lie in a more complicated interaction
of several nonlinear optical processes. We explore this
problem theoretically in the following section through
simulations.
3. THEORETICAL AND NUMERICAL
STUDIES
Based on the plane wave and slowly varying envelope approximations, the following coupled partial differential
equations can be used to describe the evolution of the fundamental and the second-harmonic pulses:
⳵A1
⳵z
⫹
冉
1
v g,1
⫺
1
v g,2
冊
⳵A1
⳵␶
⫺
j
2
␤ 2,1
⳵ 2A 1
⳵␶ 2
⫽ ⫺ j ␥ d 共 z 兲 A 2 A 1* exp关 ⫺j⌬k 0 共 z 兲 z 兴
Fig. 8. Spectral phase curve of the pump under the phasematched and high-power input condition. Measured at ⬃10 pJ
power level.
tical spectrum analyzer (see Fig. 6). For these measurements the phase-matching wavelength was set to
⬃1558.5 nm. When the SHG efficiency is low, the output
pump spectrum has a shape similar to that of the input
pump spectrum, which indicates a uniform depletion of
the pump spectral components. As the pump power increases, the pump spectrum becomes asymmetric, even
though the phase-matching wavelength is at the center of
the pump spectrum. The shorter-wavelength side is significantly suppressed, and a pronounced spectral dip
forms around 1554 nm when the pump power is close to
10 pJ, which coincides with the occurrence of the doublehump temporal shape discussed above. Simultaneously,
some spectral broadening was also observed in both the
⫺j
⳵A2
⳵z
⫺
j
2
␤ 2,2
n 2 k 1,0
n0
共 兩 A 1兩 2 ⫹ 2 兩 A 2兩 2 兲 A 1 ⫺
␣1
2
A1 ,
(1)
A2 ,
(2)
⳵ 2A 2
⳵␶ 2
⫽ ⫺ j ␥ d 共 z 兲 A 12 exp关 j⌬k 0 共 z 兲 z 兴
⫺j
n 2 k 2,0
n0
共 兩 A 2兩 2 ⫹ 2 兩 A 1兩 2 兲 A 2 ⫺
␣2
2
where A i is the envelope function of the electric field and
v g,i is the group velocity. Here the indices 1 and 2 represent the pump and the second harmonic, respectively;
␣ i is the linear loss of the material, ␤ 2,i is the GVD parameter, ␥ is the SHG coupling coefficient, n 2 is the thirdorder nonlinear refractive index, and n 0 is the refractive
Zheng et al.
Vol. 19, No. 4 / April 2002 / J. Opt. Soc. Am. B
index. For LiNbO3 , we take n 0 ⫽ 2.14; n 2 is of the order
of 10 ⫻ 10⫺16 cm2 /W for bulk lithium niobate,20 and the
resulting maximum nonlinear phase shift for a 10-pJ
pump pulse energy should be less than 0.1 rad. We ignore the variation in these numbers for different wavelengths (this does not affect our results much, as is explained below). Here ␶ ⫽ t ⫺ z/v g,2 is used, which
corresponds to a frame of reference moving with the
second-harmonic pulse at velocity v g,2 . ⌬k 0 is the phase
mismatch at the center frequencies of the pump and
second-harmonic pulses.
The above equations include the effects of GVM, linear
loss, SPM (and cross-phase modulation, XPM) and SHG,
including pump depletion. Though the GVD characteristics of the waveguide are not readily available, we take
␤ 2 ⫽ 100 ps2 /km based on the estimation from the bulk
values at the pump wavelength.10 For the secondharmonic pulse, the same ␤ 2 number was used. Because
of the relatively narrow bandwidth of the secondharmonic signal, the resulting error should be small. For
⬃200-fs pulses, the dispersion length21 is several times
larger than our sample length. This is also consistent
with our experimental observations (see Fig. 5). Thus
GVD does not seem to be significant in our experiments.
Nevertheless, it is included in the model for completeness.
Because of the GVM, the second-harmonic pulse has a
long temporal shape, and its peak power is much lower
than that of the pump pulse. Therefore it was found that
the XPM terms and the SPM term for the secondharmonic signal affect the outcomes of our simulation significantly less than the SPM term corresponding to the
pump signal. Based on the intrinsic material nonlinearity, these ␹ ( 3 ) effects should be weak within our power
range. The key parameters used in the simulation are
listed in Table 1.
In periodically poled devices, d(z) is a periodic function
of z, and the SHG phase mismatch is nonzero. In our
simulation, however, for simplicity we consider only the
first-order effect of the QPM structure. For ideal firstorder QPM, the SHG material can be considered uniform
and perfectly phase matched with an effective nonlinearity d eff ⫽ 2d/␲ (where d is the intrinsic nonlinear susceptibility). This allows us to treat d(z) as a constant
throughout the material and to set ⌬k 0 ⫽ 0. As the interacting pulses have relatively broad bandwidths, the
phase mismatch of interactions between different fre-
quency components deviates from ⌬k 0 . In Eqs. (1) and
(2), the GVM term takes into account the linear spectral
variation of the phase mismatch, though the ⌬k 0 term itself has no frequency dependence. This is equivalent to
the frequency domain equations used in Ref. 11 with no
explicit GVM term but a frequency-dependent phasemismatch term. We note that this is fundamentally different from pure frequency-domain studies of cascaded
␹ ( 2 ) processes, where the phase mismatch is taken as a
constant number. We found that other effects (discussed
below in this section) have relatively weak effects on the
calculated small-signal SHG efficiency.
In real devices the local QPM wavelength may vary because of either a nonideal device or nonuniform experimental conditions, such as oven temperature distributions. Any such deviations from uniform QPM can be
described by a spatially varying ⌬k 0 (z) term. This effect
is further discussed below.
Equations (1) and (2) were solved numerically by a
symmetrized split-step Fourier method.11,23,24 The parameters used in the simulation, such as waveguide loss
and the GVM parameter, were decided on the basis of either previously published values4,25 or values near known
bulk values within a reasonable range.
We first calculated the SHG efficiency for an input
sech(t/T0) pulse with T 0 ⫽ 140 fs (corresponds to an intensity profile with a duration of 246 fs FWHM), with the
GVD and SPM terms set to zero. The results represent
the ideal SHG case under large GVM, shown in Fig. 2.
Our experimental results for the efficiency clearly cannot
be explained by this simplified picture. A better match to
the results, would require more factors to be taken into
consideration.
Since our observed SHG spectra show a shape significantly different from a sinc2 shape, the phase-matching
condition may be spatially nonuniform. In our experimental setup, the end surfaces of the crystal in our homemade oven were exposed to open air. Therefore the temperatures at the end surfaces are expected to be lower
than at the center part of the oven. At high temperatures this difference could be as large as several degrees.
We try to model the temperature distribution by incorporating a varying temperature profile ⌬T(z), which results
in a spatial variation18 of ⌬k. The temperature profile
⌬T ⫽ ⌬T 1 ⫹ ⌬T 2 is taken as18
Table 1. Simulation Parameters
a
Parameter
Value
Fundamental mode size
Second-harmonic mode size
Overlap areaa
Crystal length
Refractive index
1/v g,1 ⫺ 1/v g,2
␤2
␣1
␣2
d eff
40 ␮m2
10 ␮m2
53.1 ␮m2
6 cm
2.14
0.37 ps/mm
100 ps2/km
0.35 dB/cm
0.7 dB/cm
16.5 pm/V
Refs. 12, 22.
843
⌬T i ⫽
冦
T Ni
0
T Ni
冉
z ui ⫺ z
冉
z ⫺ z oi
z Ni
z Ni
冊
冊
pi
z ⭐ z ui
z ui ⬍ z ⭐ z oi .
pi
(3)
z oi ⬍ z
We adjusted the parameters in this equation to try
to match the low-power SHG spectrum.
For z o1
⫽ 28.75 mm, z u1 ⫽ 26.75 mm, z N1 ⫽ 10 mm, p 1 ⫽ 4.3,
T N1 ⫽ 0.06, z o2 ⫽ 27.75 mm, z u2 ⫽ 27.75 mm, z N2
⫽ 25 mm, p 2 ⫽ 1.2, and T N2 ⫽ 0.72, the simulated SHG
spectrum is given in Fig. 9. With these parameters the
effective temperature at the ends is ⬃6 deg lower than at
the center. The result shows asymmetric and oscillatory
844
J. Opt. Soc. Am. B / Vol. 19, No. 4 / April 2002
Fig. 9. Simulated and experimental second-harmonic spectrum
under low input power. Solid curve, simulation result; Dashed
curve, experimental result.
Fig. 10. Simulated pump spectrum under 10 pJ of input power,
considering an effective varying phase mismatch and an effective
NPM term.
Zheng et al.
To reasonably fit the simulation results to the data,
we found that it was necessary to include a nonlinear phase-modulation (NPM) term for the pump pulse
with a large negative coefficient (approximately ⫺2000
⫻ 10⫺16 cm2 /W). This term uses the same form as the
SPM term in Eq. (1) but with an assumed nonlinear coefficient that is negative and ⬃2 orders of magnitude larger
than the intrinsic n 2 . When the large NPM term is included, the calculated pump spectrum shows a shortwavelength dip and a phase jump similar to what we
measured in the experiments (see Fig. 10). The simulated temporal intensity profile at a 10-pJ pump power,
shown in Fig. 11, is double peaked, again similar to our
experimental finding. Such important features are absent in other simulations that include the spatially varying ⌬k term but only the intrinsic n 2 . The simulated
SHG efficiency (Fig. 12) also matches the measured results much more closely. The combination of a spatially
varying ⌬k and the NPM term yield a significantly closer
match to the experimental results at our highest-power
conditions, though it is still difficult to match all the results at all power levels.
We also simulated our experimental studies (see Fig. 7)
in which the input center wavelength was detuned from
the phase-matching wavelength. A 10-pJ input energy
was assumed. Different constant phase mismatches
were introduced in the simulation in order to change the
relative difference between the input wavelength and the
phase-matching wavelength. This is the same as experimentally tuning the input wavelength while keeping the
phase-matching wavelength constant. As is shown in
Fig. 13, the simulation results are similar to the experimental data. For the cases that correspond to tuning to
longer input wavelengths by a few nanometers, there is
not much spectral distortion. On the other hand, for tuning to shorter input wavelengths, the simulated results
resemble the experimental results qualitatively, although
the extension of the simulated spectra on the shortwavelength side is somewhat too large.
The picture so far is as follows. First, the large NPM
generates a chirp in the pump pulse. Second, because of
the large temporal walk-off, the trailing edge of the pump
interacts differently with the second-harmonic signal
than does the front edge of the pulse. The observed spectral and temporal distortions arise because of the separation of different pump frequency components in the time
Fig. 11. Simulated pump pulse shape under 10 pJ of input
power, considering an effective varying phase mismatch and an
effective NPM term.
features similar to the measured result. As is shown in
Fig. 12 below, the spatially varying phase mismatch leads
to some reduction in the SHG efficiency at high powers
but still does not account for the bulk of the saturation.
We also found that inclusion of the spatially varying
phase mismatch is insufficient to explain our observations
of strong temporal and spectral reshaping of the pump
pulse.
Fig. 12. Simulated (Simu) second-harmonic conversion efficiency versus input pump power, considering the effects of different physical processes.
Zheng et al.
Fig. 13. Simulated pump spectrum under high-power input condition and varied phase-matching conditions. The legend shows
the difference between the input pump center wavelength and
the phase-matching wavelength.
domain coupled with their interaction with the secondharmonic wave, which also walks off from the pump pulse
in time.
Note that for most materials the intrinsic n 2 from the
optical Kerr effect is positive and much smaller than what
we assumed in our simulation, except in the presence of
nonlinear absorptions.26 In the following we discuss the
physical origin of the large negative NPM, which we have
included phenomenologically to better match the simulation results with experimental observations.
One possibility that has been discussed is that higher
Fourier components of the QPM grating may result in an
intensity-dependent phase shift.27,28 The idea is that
considering only the lowest Fourier component of the
QPM structure does not give complete results when large
growth of the second-harmonic wave occurs within a
single QPM period. When significant pump depletion occurs in a single period, as in Ref. 28 for intensities of tens
of giga watts per square centimeter and higher in bulk
PPLN, the local phase mismatch within a single QPM period must also be considered. At our relatively low pump
power levels (⬍1 GW/cm2), substantial growth of the
second-harmonic wave requires of the order of a few hundred QPM periods. Therefore considering only the lowest Fourier component of the QPM grating should be
valid. We conclude, then, that the small, positive phase
shift, as described in Ref. 26, cannot explain the large,
negative NPM that we are considering here.
The more plausible explanation of the NPM may come
from cascaded ␹ ( 2 ) processes. In a parametric ␹ ( 2 ) process with nonzero phase mismatch, the conversion and
backconversion between the waves can add phase shifts
caused by the phase mismatch of the interacting waves.
These phase shifts are said to constitute a cascaded ␹ ( 2 )
process, which has been shown to resemble a ␹ ( 3 )
process.29–32 The most common cascaded ␹ ( 2 ) effects involve a single-step process, such as SHG. Cascade phase
shifts in multistep parametric processes, such as thirdharmonic generation (THG) arising from SHG followed by
sum-frequency generation, have also been discussed.33–35
Such cascade phenomena have been intensively studied as substitutes for intrinsic ␹ ( 3 ) effects for applications
like all-optical switching36 and pulse compression.37 An
important characteristic of the cascaded ␹ ( 2 ) process is
Vol. 19, No. 4 / April 2002 / J. Opt. Soc. Am. B
845
that the magnitude and polarity of the phase shift are related to the phase mismatch of the ␹ ( 2 ) process. Nonlinear phase shifts, which are very large compared with intrinsic ␹ ( 3 ) processes, as well as both positive and
negative effective n 2 , 36 can be realized by controlling the
phase-mismatch condition. It has been shown that an effective n 2 as large as ⬃10 ⫻ 10⫺12 cm2 /W can be obtained
from SHG processes in PPLN.36
Theoretically, Eqs. (1) and (2) already include the possibility of cascaded ␹ ( 2 ) phase shifts within the main SHG
process. The simulations show that any such phase
shifts are far too small to explain our experimental observations. However, the additional large NPM may originate from cascade effects in an additional, simultaneous
parametric process. Although single cascaded processes
have been studied extensively, the interaction between a
phase-matched parametric process and an additional cascaded process remains largely unexplored, especially experimentally. Our results here suggest that such effects
can have a significant impact on real applications.
The additional cascaded process could come from either
another SHG channel, as we describe below, or a multistep process. Experimentally we observed some light
from a green generation process. Simultaneous occurrence of more than one QPM ␹ ( 2 ) process has been observed often in QPM materials.17,38,39 For PPLN, under
our phase-matching condition, a third-order QPM green
generation process based on sum-frequency generation
can also be nearly phase matched. Such a THG process
has been shown in Refs. 33–35 potentially to generate
nonlinear phase shifts. Another possibility is an optical
parametric amplification process that involves the fundamental pump mode, the generated second-harmonic signal, and another higher-order spatial mode of the pump.
As noted above, our LiNbO3 waveguide devices support
multiple transverse spatial modes at both the pump and
SHG wavelengths. Because different spatial modes have
different effective refractive indices, different pairs of fundamental and second-harmonic modes phase match at
different wavelengths.12 Because of the mode filter, only
the fundamental spatial mode of the pump is initially
seeded; however, higher-order spatial modes at the pump
wavelength can be generated through a differencefrequency generation process. We have performed simulations of both the THG and optical parametric amplification multistep processes. At the relatively low power
levels investigated here, our simulations suggest that the
influence from these processes is relatively weak. For example, in the THG case, the strength of the sumfrequency mixing between fundamental and second harmonic wavelengths must be artificially increased by more
than an order of magnitude from the normally acceptable
level to yield the nonlinear phase shifts needed to give a
small second peak in the pump spectrum at 10 pJ of input
energy. Also, the NPM caused by such multistep cascaded processes behaves more like a higher-order
nonlinearity33 than like SPM. As a result, our simulation results based on these multistep models vary much
more rapidly with changes in pump power than do our experimental results.
From our simulations with different models, it appears
that a cascaded SHG process involving a second (higher-
846
J. Opt. Soc. Am. B / Vol. 19, No. 4 / April 2002
Zheng et al.
order) spatial mode of the second harmonic could be the
most likely source of the large NPM. For a fixed temperature, the phase-matching wavelengths for either the
fundamental or a higher-order second-harmonic mode
could be several nanometers to tens of nanometers apart.
To model the effect of an additional nearly phase-matched
SHG channel, we used a modified set of SHG equations:
⳵A1
⳵z
⫹
冉
1
v g,1
⫺
1
v g,2
冊
⳵A1
⳵␶
⫽ ⫺j ␥ dA 2 A 1* exp共 ⫺j⌬k 0 z 兲
⫺j ␥ ⬘ dA 2⬘ A 1* exp共 ⫺j⌬k 0⬘ z 兲
⫺
⳵A2
⳵z
␣1
2
A1 ,
⫽ ⫺j ␥ dA 12
⫻ exp共 j⌬k 0 z 兲 ⫺
⳵ A 2⬘
⳵z
(4)
␣2
2
Fig. 16. Simulated pump pulse shape under a 10-pJ pump
power level considering a second phase-mismatched SHG process.
A2 ,
(5)
A2⬘ ,
(6)
⫽ ⫺j ␥ ⬘ dA 12
⫻ exp共 j⌬k 0⬘ z 兲 ⫺
␣2
2
Fig. 17. Simulated pump spectrum and spectral phase curve
under a 10-pJ pump power level considering a second phasemismatched SHG process, assuming a phase mismatch of sign
opposite that assumed in Fig. 16.
Fig. 14. Simulated pump spectrum under varied pump power
conditions considering a second phase-mismatched SHG process.
Fig. 15. Simulated pump spectrum and spectral phase curve
under a 10-pJ pump power level considering a second phasemismatched SHG process.
where Eq. (6) and the terms involving A 2⬘ represent the
phase-mismatched SHG channel that could generate the
nonlinear phase shift. We assumed that the group velocity and losses of the two second-harmonic signals are the
same and set the GVD and ␹ ( 3 ) SPM and XPM terms to
zero. We use the same nonlinear coupling coefficient for
the two SHG process ( ␥ ⫽ ␥ ⬘ ). This ignores the different overlap integrals between the pump mode and the two
SHG modes, but this should be sufficient to gain physical
insight. The phase-matching wavelength of the second
SHG process is taken as 15 nm longer than the center
wavelength.
Figure 14 shows the pump spectrum under different
pump power levels. The results show the generation of
the spectral dip and larger sidelobes at pulse energies
above 2.5 pJ. Compared with Fig. 6, the simulation results have strong similarities to the experimental results
for comparable power levels. It is noted that the simulated spectral distortions seem to be slightly larger than
those experimentally observed. This could indicate that
the strength of the cascaded ␹ ( 2 ) process that we used is
somewhat larger than the real one. Shown in Fig. 15 is
Zheng et al.
the simulation results of the spectral amplitude and
phase at a 10-pJ pump energy. The magnitude and
shape of the spectral phase jump around the spectral dip
also matches that in Fig. 8. In the temporal domain, the
breakup of the pump pulse is shown in the simulation in
Fig. 16. Thus these results match the experimental results and the earlier simulation results reasonably well
and also prove that a cascaded ␹ ( 2 ) process significantly
away from the center, wavelength could generate strong
nonlinear phase distortions under certain conditions.
As in other cascaded ␹ ( 2 ) processes, the sign of the
phase mismatch determines the sign of the nonlinear
phase shift. This in turn decides the sense of the distortion in the pump spectrum. As is shown in Fig. 17, when
we assume that the second SHG channel is phase
matched at 1545 nm, the simulated output pump spectrum shows a spectral dip on the longer-wavelength side,
as opposed to the short-wavelength dip seen in Fig. 14.
We note that since there may be several waveguide
modes, there is also the possibility of having more than
one important additional cascaded ␹ ( 2 ) process. Although the physical picture would become even more complicated, the results described above should still be relevant.
4. CONCLUSION
In conclusion, we have investigated high-power femtosecond SHG under large GVM conditions. It appears that a
very strong NPM may contribute to a stronger-thanexpected saturation of the SHG conversion efficiency at
an approximately 60% level and to a distinct temporal
and spectral distortion of the pump pulse. Our simulation results based on cascaded ␹ ( 2 ) effects involving SHG
to a second higher-order SH mode give a good qualitative
and, in some cases, quantitative match to our experimental results, especially in light of the uncertainties in our
model.
We believe we may have observed an interesting case of
the interaction between a phase-matched SHG process
and a second phase-mismatched SHG process, which results in a cascaded ␹ ( 2 ) process, significantly changing the
SHG yield. Such phenomena have not been reported previously to our knowledge and may have interesting scientific and practical significance. One of the most attractive features of QPM devices is the potential to permit one
to engineer the nonlinearities. Our studies suggest that,
for some applications, simultaneously controlling several
waveguide modes may be necessary. This could become
both a challenge and an opportunity to realize nonlinear
optical waveguide devices with novel functionalities.
Vol. 19, No. 4 / April 2002 / J. Opt. Soc. Am. B
Z. Zheng’s e-mail address is zzheng4@lucent.com.
*Present address, Lucent Technologies, Holmdel,
N.J. 07733-3030.
†
Present address, Tellium Inc., Oceanport, N.J. 077570901.
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ACKNOWLEDGMENTS
The work at Purdue University was supported by the National Science Foundation under grant 9900369-ECS.
The work at Stanford University was supported by the
Defense Advanced Research Projects Agency through the
Optoelectronic Materials Center. The authors acknowledge helpful discussions with Gennady Imeshev.
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